Superconvergence of Legendre projection methods for the eigenvalue problem of a compact integral operator

摘要

In this paper, we consider the Galerkin and collocation methods for the eigenvalue problem of a compact integral operator with a smooth kernel using the Legendre polynomials of degree ≤n. We prove that the error bounds for eigenvalues are of the order O(n-2r) and the gap between the spectral subspaces are of the orders O(n-r) in L2-norm and O(n12-r) in the infinity norm, where r denotes the smoothness of the kernel. By iterating the eigenvectors we show that the iterated eigenvectors converge with the orders of convergence O(n-2r) in both L2-norm and infinity norm. We illustrate our results with numerical examples.